We thoroughly study the semantics of inductive and recursive types. Our point of view is that types constitute the objects of a category and that type constructors are bifunctors on the category of types. By a bifunctor on a category we mean a functor on two variables from the category to itself, contravariant in the first, covariant in the second.

First, following Peter Freyd, the stress is on the study of algebraically complete categories, i.e. those categories admitting all inductive types (in the sense that every endofunctor on them has an initial algebra—this is understood in a setting in which the phrase “every endofunctor” refers to a class of enriched endofunctors—see Definition 6.1.4). After observing that algebraic completeness guarantees the existence of parameterised initial algebras, we identify, under the name of parameterised algebraically complete categories, all those categories which are algebraically complete and such that every parameterised inductive type constructor gives rise to a parameterised inductive type (see Definition 6.1.7). Type constructors on several variables are dealt with by Bekič's Lemma, from which follow both the Product Theorem for Parameterised Algebraically Complete Categories (Theorem 6.1.14) and also the dinaturality of Fix (the functor delivering initial algebras).

Second, again following Peter Freyd, algebraic completeness is refined to algebraic compactness by imposing the axiom that, for every endofunctor, the inverse of an initial algebra is a final coalgebra. The compactness axiom is motivated with a simple argument showing that every bifunctor on an algebraically compact category admits a fixed-point.